If Union closed sets conjecture hold for all finite families, does that imply that it holds for families with some sets repeating twice (no more than that) also? If not, do we have any counter-example?
2026-02-22 19:31:33.1771788693
Special case of the Union closed sets conjecture
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Consider the family of all $8$ subsets of $\{1,2,3\}$. It is union-closed i.e. union of any two sets in the family is also in the family. Each element is contained in $4$ sets. Then let's add one copy of sets $\{1\}$, $\{2\}$, $\{3\}$. Now there are $3 + 8 = 11$ sets and each element is contained in $1+4 = 5 < {11 \over 2}$ of them. So this is not true for the families with duplicates.